Method for performing HARQ by using polar code having random length

ABSTRACT

The present specification provides a method for performing a physical layer security-based hybrid automatic repeat request (HARQ). The method comprises the steps of: generating a first code comprising information bits for forming data to be transmitted, and non-information bits which are unrelated to the data to be transmitted; generating a second code, having a length which differs from that of the first code, by puncturing the first code; determining locations of the information bits and non-information bits within the second code such that a variance of mutual information for each of the information bits and the non-information bits is maximized; and performing the HARQ by using the second code in which the locations of the information bits and the non-information bits are determined.

This application is a 35 USC § 371 National Stage entry of InternationalApplication No. PCT/KR2016/014411, filed on Dec. 9, 2016, and claimspriority to U.S. Provisional Application No. 62/387,589, filed on Dec.24, 2015, all of which are incorporated by reference in their entiretyherein.

BACKGROUND OF THE INVENTION Field of the invention

The present invention relates to wireless communications. Moreparticularly, the present invention relates to secure communicationsusing polar codes.

Related Art

It is important to transfer data without error from a transmitter to areceiver in a data communication system. In 1948, Shannon mathematicallyinvestigated a limitation of a maximum data transfer rate at which datacan be transferred without error, which is called channel capacity. Inorder to implement a real communication system close to the channelcapacity, an error correction code having implementable complexity mustexist. Several types of error correction codes have been developed since1948, and turbo codes and low density parity check (LDPC) or the likehave been developed relatively recently as error correction codes whichexhibit performance close to channel capacity of Shannon. However,although these codes exhibit performance close to the channel capacityof Shannon, accurate channel capacity is not achieved. A polar code hasrecently been developed as a code which completely satisfies the channelcapacity mathematically while satisfying such a problem.

Meanwhile, one of important methods for transmitting data without errorby using an error correction code is Hybrid Automatic Repeat reQuest(HARQ). In the conventional Automatic Repeat reQuest (ARQ) scheme, whena receiver receives a packet transmitted by a transmitter without error,the receiver transmits ACK through a feedback channel, and thetransmitter transmits a next packet for carrying new data. However, ifthere is an error in the packet received by the receiver, the receiverdiscards the erroneous packet and transmits NACK through the feedbackchannel, and the transmitter retransmits the previously transmittedpacket.

HARQ which is evolved from the existing ARQ scheme is a scheme in whichtransmission data is transmitted after coding it by using an errorcorrection code when the transmitter transmits data. When NACK isreceived from the receiver through the feedback channel, the transmittermay retransmit the same packet as the previously transmitted packet ormay transmit only new code bits. Instead of discarding the erroneouspacket, the receiver effectively decodes data by combining informationin the erroneous packet and information in the newly received packet.The HARQ scheme may be performed in combination with several errorcorrection codes.

In addition to transferring of data from a transmitting end to areceiving end without error through a channel, it is important in acommunication system to allow only authenticated receivers to decode thetransmitted data and not to allow other unauthenticated receivers todecode the data. Such a communication security problem has beenconventionally solved by sharing the same security key between thetransmitter and the receiver according to cryptography. Forcommunication security based on the cryptography, a sufficiently strongsecurity key must be generated, the generated security key must besecurely distributed to the transmitter/receiver, and the security keymust be periodically updated and managed. However, in general, a methodof generating, distributing, and managing the security key is notsimple. In particular, it is not easy for several types of wirelessnetworks to reliably perform communication by generating anddistributing the security key within a short time.

In order to solve this problem, a new approach has been studied toachieve communication security, which is called physical layer security.The physical layer security achieves communication security at aphysical layer, not at an upper layer as in cryptography. One of themost effective ways to provide substantial communication security basedon the physical layer security is to use a polar code.

Recently, studies on the polar code have been actively conducted inacademia and industry. One topic of them is to realize a polar codehaving a random code length. Basically, a length of the polar code maybe given as a square of 2. For example, a polar code having a length of2⁸=256 or 2⁹=512 can be realized, but a polar code having a randomlength longer than 256 and shorter than 512 cannot be generated in anoriginal construction manner. In addition, although several schemes havebeen proposed to realize the polar code having the random length, it isdifficult to consider them as optimal polar code construction schemessince these schemes are not optimized in terms of mutual informationwhich is an ultimate performance indicator of the communication system.In addition, it is difficult to apply the existing schemes forconstructing the polar code having the random length to physical layercommunication.

Meanwhile, various studies have been conducted on a polar coding-basedHARQ scheme which combines polar coding and HARQ. However, according tothe schemes proposed up to now, the polar code has not been constructedto improve information of channel polarization which is basic concept ofpolar coding. The schemes proposed up to now have a problem in that aninformation bit to be retransmitted is not coded when the informationbit is retransmitted based on a repetition coding scheme. In addition,the schemes proposed up to now have a limitation in that performance ofthe polar coding-based HARQ is not optimized in terms of mutualinformation, and has not been developed for physical layer securitycommunication.

SUMMARY OF THE INVENTION

A disclosure of the present specification aims to provide a method ofgenerating a polar code having a random length optimized in terms ofmutual information so as to be suitable to physical layer securitycommunication.

Another disclosure of the present specification aims to provide a methodof performing a hybrid automatic repeat request (HARQ) by using a polarcode having a random length optimized in terms of mutual information.

To achieve the aforementioned purpose, a disclosure of the presentspecification provides a method of performing a HARQ based on physicallayer security. The method may include: generating a second code havinga length different from that of a first code by puncturing the firstcode including an information bit constituting data to be transmittedand a non-information bit irrelevant to the data to be transmitted;determining locations of the information bit and non-information bit inthe second code; applying the determined locations of the informationbit and non-information bit determined based on the second code to thefirst code; and performing the HARQ by using a packet generated bysplitting the first code.

In the generating of the second code, mutual information may becalculated based on a probability distribution of a log likelihood ratio(LLR) for the first code, and the first code is punctured to decrease aloss of the calculated mutual information. In this case, the probabilitydistribution of the LLR may be calculated by using Gaussianapproximation.

In the determining of the locations of the information bit and thenon-information bit, the locations of the information bit and thenon-information bit may be determined to minimize a variance of mutualinformation for each of the information bit and the non-information bit.

In addition, a ratio of the information bit and non-information bitincluded in the second code may be determined by an indication signalpreviously received.

In order to achieve the aforementioned purpose, another disclosure ofthe present specification provides an apparatus for performing HARQbased on physical layer security. The apparatus may include: a radiofrequency (RF) unit transmitting and receiving a radio signal; and aprocessor controlling the RF unit. The processor may be configured to:generate a second code having a length different from that of a firstcode by puncturing the first code including an information bitconstituting data to be transmitted and a non-information bit irrelevantto the data to be transmitted; determine locations of the informationbit and non-information bit in the second code; apply the determinedlocations of the information bit and non-information bit determinedbased on the second code to the first code; and perform the HARQ byusing a packet generated by splitting the first code.

According to a disclosure of the present specification, it is possibleto construct a polar code having a random length optimized in terms ofmutual information. In particular, it is possible to decrease acalculation amount required to construct the polar code having therandom length by using a suboptimal code design scheme. In addition, itis possible to decrease a communication error and maintain communicationsecurity by extendedly applying the polar code having the random lengthto physical layer security communication.

In addition, it is possible to decrease a communication error andmaintain communication security while decreasing a calculation amountrequired in HARQ, by performing the HARQ on the basis of the polar codehaving the random length optimized in terms of the mutual information.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an example of a mother code having a length of 8 and apunctured code having a length of 6.

FIG. 2 shows an example of four cases considered when polar codesub-optimization is performed when N=8, M=6, and Δ=2.

FIG. 3 shows cases considered when optimization of a polar code isperformed and not considered when sub-optimization is performed.

FIG. 4 shows two cases where mutual information is calculated in polarcode sub-optimization when N=8, M=4, and Δ=4.

FIG. 5 shows an example of six cases where mutual information iscalculated in polar code sub-optimization when N=8, M=4, and Δ=2.

FIG. 6 shows an example of a simplified bit grouping algorithm for anHARQ scheme 1.

FIG. 7 is a conceptual diagram for explaining concept of a simplifiedbit grouping algorithm for an HARQ scheme 1.

FIG. 8 shows an example of transmission based on an HARQ scheme 2A.

FIG. 9 shows an example of transmission based on an HARQ scheme 2B.

FIG. 10 shows an example of transmission according to an HARQ scheme 2C.

FIG. 11 shows an example of comparing an HARQ scheme 2A, an HARQ scheme2B, and an HARQ scheme 2C.

FIG. 12 shows an example of a scheme in which an HARQ scheme 1 and anHARQ scheme 2 are combined.

FIG. 13 shows another example of a scheme in which an HARQ scheme 1 andan HARQ scheme are combined.

FIG. 14 shows an example of values i_({circumflex over (b)}) and i_(ê)for a mother code.

FIG. 15 shows an example of transmission according to a simplifiedsecure HARQ scheme 1A.

FIG. 16 is a conceptual view for explaining concept of a simplifiedsecure HARQ scheme 1A.

FIG. 17 shows an example of transmission according to a simplifiedsecure HARQ scheme 1B.

FIG. 18 is a conceptual view for explaining concept of a simplifiedsecure HARQ scheme 1B.

FIG. 19 is a flowchart showing a method of performing HARQ by using apolar code having a random length according to an embodiment of thepresent invention.

FIG. 20 is a block diagram illustrating a wireless communication systemin which the present disclosure is implemented.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

The technical terms used herein are used to merely describe specificembodiments and should not be construed as limiting the presentinvention. Further, the technical terms used herein should be, unlessdefined otherwise, interpreted as having meanings generally understoodby those skilled in the art but not too broadly or too narrowly.Further, the technical terms used herein, which are determined not toexactly represent the spirit of the invention, should be replaced by orunderstood by such technical terms as being able to be exactlyunderstood by those skilled in the art. Further, the general terms usedherein should be interpreted in the context as defined in thedictionary, but not in an excessively narrowed manner.

The expression of the singular number in the specification includes themeaning of the plural number unless the meaning of the singular numberis definitely different from that of the plural number in the context.In the following description, the term ‘include’ or ‘have’ may representthe existence of a feature, a number, a step, an operation, a component,a part or the combination thereof described in the specification, andmay not exclude the existence or addition of another feature, anothernumber, another step, another operation, another component, another partor the combination thereof.

The terms ‘first’ and ‘second’ are used for the purpose of explanationabout various components, and the components are not limited to theterms ‘first’ and ‘second’. The terms ‘first’ and ‘second’ are only usedto distinguish one component from another component. For example, afirst component may be named as a second component without deviatingfrom the scope of the present invention.

It will be understood that when an element or layer is referred to asbeing “connected to” or “coupled to” another element or layer, it can bedirectly connected or coupled to the other element or layer orintervening elements or layers may be present. In contrast, when anelement is referred to as being “directly connected to” or “directlycoupled to” another element or layer, there are no intervening elementsor layers present.

Hereinafter, exemplary embodiments of the present invention will bedescribed in greater detail with reference to the accompanying drawings.In describing the present invention, for ease of understanding, the samereference numerals are used to denote the same components throughout thedrawings, and repetitive description on the same components will beomitted. Detailed description on well-known arts which are determined tomake the gist of the invention unclear will be omitted. The accompanyingdrawings are provided to merely make the spirit of the invention readilyunderstood, but not should be intended to be limiting of the invention.It should be understood that the spirit of the invention may be expandedto its modifications, replacements or equivalents in addition to what isshown in the drawings.

Hereinafter, the present specification proposes two schemes related topolar coding for transmitting data without error. First, it is proposeda scheme for optimally constructing a polar code having a random length.In addition, it is proposed to extendedly apply a polar code having alimited random length to physical layer security communication. Second,it is proposed an HARQ scheme based on polar coding capable ofeffectively transmitting data by using the constructed polar code. Inaddition, it is proposed a scheme of extendedly applying the proposedpolar coding-based HARQ scheme to the physical layer securitycommunication.

1. Scheme of Constructing Polar Code Having Random Length

First, it is considered a case of constructing a polar code having arandom length M. To construct the polar code having the length M, amother code having a length N=2^(n) may be punctured. In this case, ndenotes a natural number, and N and M are related as follows.

$\begin{matrix}{\frac{N}{2} = {{2^{n - 1} < M < N} = 2^{n}}} & \left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack\end{matrix}$

If T_(M) denotes a set of indices of M symbols before being subjected tocoding without puncturing, the following relationship is satisfied.

_(M)⊂

_(N)={1,2, . . . N}  [Equation 2]

In addition, if a code R_(M) denotes an index set of M symbols to betransmitted after being subjected to coding without puncturing, R_(M) iseventually equal to a set of indices of symbols received in a receivingend, and the following relationship is satisfied.

_(M)⊂

_(N)={1,2, . . . ,N}  [Equation 3]

FIG. 1 shows an example of a mother code having a length of 8 and apunctured code having a length of 6.

One example of constructing a polar coding having a length of M=6 from apolar code having a length of N=8 is shown in FIG. 1. An index set T_(N)of all symbols of the mother code is given by T_(N)={1, 2, 3, 4, 5, 6,7, 8}. A set of 6 symbols before being subjected to coding withoutpuncturing is given by {μ₁, μ₃, μ₄, μ₆, μ₇, μ₈}, and thus T_(M) is givenby T_(M)={1, 3, 4, 6, 7, 8}. In addition, a set of symbols to betransmitted after being subjected to coding without puncturing is givenby {X₂, X₃, X₄, X₅, X₆, X₈}, and a set of coded symbols to be receivedin a receiving end is given by {Y₂, Y₃, Y₄, Y₅, Y₆, Y₈}. Therefore, aset R_(M) is given by R_(M)={2, 3, 4, 5, 6, 8}.

The construction of the polar code having the length of M is todetermine the aforementioned T_(M) and R_(M) in an optimal manner. Upondetermining the sets T_(M) and R_(M), M rows corresponding to T_(M) isselected from an N×N polar code generation matrix G_(N) of the mothercode, and an M×M polar code generation matrix (G_(N))

is constructed.

1.1 Optimization of Polar Code Having Random Length

When constructing a code having a length M by puncturing (N−M) symbolsfrom a mother code having a length N, there is a loss in mutualinformation. A code may be constructed through the followingoptimization so that the constructed length M has optimal performance interms of the mutual information by minimizing the loss in the mutualinformation.

$\begin{matrix}{{\left( {{\hat{\mathcal{T}}}_{M},{\hat{\mathcal{R}}}_{M}} \right) = {\arg\;{\max\limits_{\mathcal{T}_{M},\mathcal{R}_{M}}\;{I\left( {U_{\mathcal{T}_{M}};Y_{\mathcal{R}_{M}}} \right)}}}},\mathcal{T}_{M},{\mathcal{R}_{M} \in \Psi_{N}^{M}}} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$

Herein, it is given by U_(T) _(M) ={U_(i):i∈

_(M)} and Y_(R) _(M) ={Y_(i):i∈

_(M)}. That is, I(U_(T) _(M) ; T_(R) _(M) ) denotes mutual informationof a generated polar code having a length M. In Equation 4, Ψ_(N) ^(M)denotes a set of all subsets having M elements of T_(N)={1, 2, . . . ,N}. That is, Ψ_(N) ^(M) is given as follows.Ψ_(N) ^(M)={{1,2, . . . ,M},{2,3, . . . ,M+1}, . . . ,{1,3,4, . . .,M+1}, . . . }  [Equation 5]

Therefore, the number of all subsets having M elements included in Ψ_(N)^(M) is given by

$\begin{pmatrix}N \\M\end{pmatrix} = {\frac{N!}{{M!}{\left( {N - M} \right)!}}.}$

However, optimization of the code by directly using the Equation 4 aboveis difficult due to the following reasons.

First, it is not easy to directly calculate the mutual information.

Second, although complexity of optimization using Equation 4 is, N and Mare substantially great number. Therefore, it is difficult to performsubstantial optimization.

Therefore, the present specification proposes a method of constructingan optimized polar code by solving the aforementioned problem.

First, the following result can be mathematically proved.

_(M)=

_(M)  [Equation 6]

That is, when T_(M) and R_(M) are given identically, the Equation 4above is optimized (that is, mutual information is maximized).

By using such a result, the optimization of the Equation 4 above may besimplified as follows.

$\begin{matrix}{{{\overset{\hat{}}{\mathcal{T}}}_{M} = {\arg{\max\limits_{\mathcal{T}_{M}}{I\left( {U_{\mathcal{T}_{M}};\ Y_{\mathcal{T}_{M}}} \right)}}}},{\mathcal{T}_{M} \in \Psi_{N}^{M}}} & \left\lbrack {{Equation}\mspace{11mu} 7} \right\rbrack\end{matrix}$

As such, complexity of simplified optimization is

$\begin{pmatrix}N \\M\end{pmatrix},$and may be significantly lower than optimization complexity

$\begin{pmatrix}N \\M\end{pmatrix} \times \begin{pmatrix}N \\M\end{pmatrix}$based on the previous Equation 4.

However, the optimization based on the Equation 7 above still has thefollowing two problems.

First, it is not easy to directly calculate the mutual information

Second, although complexity of the optimization using Equation 7 is, Nand M are substantially great number. Therefore, it is still difficultto perform substantial optimization.

Hereinafter, a method of calculating the mutual information is proposedto solve the first problem. First, a log likelihood ratio (LLR) L_(i)(·)for a current input symbol U_(i) is defined. In a polar code, the LLR isgiven as a function of all outputs Y_(l) ^(N) and previous inputs U_(l)^(i-1). Since the LLR has sufficient statistics for U_(i), total mutualinformation in a polar code having a length N may be as follows.

$\begin{matrix}\begin{matrix}{{I\left( {U_{\mathcal{T}_{N}};\ Y_{\mathcal{T}_{N}}} \right)} = {\sum\limits_{i = 1}^{N}{I\left( {U_{i};\left. Y_{N} \middle| U_{1}^{i - 1} \right.} \right)}}} \\{= {\sum\limits_{i = 1}^{N}{I\left( {U_{i};{\mathcal{L}\left( Y_{T_{N}} \middle| U_{1}^{i - 1} \right)}} \right)}}} \\{= {\sum\limits_{i = 1}^{N}{I\left( {U_{i};{\mathcal{L}_{i}\left( \mathcal{T}_{N} \right)}} \right)}}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 8} \right\rbrack\end{matrix}$

Herein, L_(i)(T_(N)) may be defined as follows.

_(i)(

_(N)):=

(Y _(T) _(N) |U ₁ ^(i−1))  [Equation 9]

Similarly, the mutual information in the polar code having the length Mmay be as follows.

$\begin{matrix}{{I\left( {U_{\mathcal{T}_{M}};Y_{\mathcal{T}_{M}}} \right)} = {\sum\limits_{i = 1}^{M}{I\left( {U_{i};{\mathcal{L}_{i}\left( \mathcal{T}_{M} \right)}} \right)}}} & \left\lbrack {{Equation}\mspace{14mu} 10} \right\rbrack\end{matrix}$

Herein, L_(i)(T_(M)) may be defined as follows.

_(i)(

_(M))=

(Y _(T) _(M) |U ₁ ^(i−1))  [Equation 11]

Therefore, the optimization based on the Equation 7 above may beexpressed as follows.

$\begin{matrix}{{{\hat{\mathcal{T}}}_{M} = {\arg\;{\max\limits_{\mathcal{T}_{M}}{\sum\limits_{i \in \mathcal{T}_{M}}{I\left( {U_{i};{\mathcal{L}_{i}\left( \mathcal{T}_{M} \right)}} \right)}}}}},{\mathcal{T}_{M} \in \Psi_{N}^{M}}} & \left\lbrack {{Equation}\mspace{14mu} 12} \right\rbrack\end{matrix}$

An average value M_(I)(T_(M)) of the mutual information may be used toexpress the optimization of Equation 11 as follows.

$\begin{matrix}{{{\overset{\hat{}}{T}}_{M} = {\arg{\max\limits_{\mathcal{T}_{M}}{M_{I}\left( \mathcal{T}_{M} \right)}}}},{\mathcal{T}_{M} \in \Psi_{N}^{M}}} & \left\lbrack {{Equation}\mspace{14mu} 13} \right\rbrack\end{matrix}$

Herein, M_(I)(T_(M)) is as follows.

$\begin{matrix}{{{M_{I}\left( \mathcal{T}_{M} \right)} = \frac{1}{M}},{\sum\limits_{i \in \mathcal{T}_{M}}{I\left( {U_{i};{\mathcal{L}_{i}\left( \mathcal{T}_{M} \right)}} \right)}}} & \left\lbrack {{Equation}\mspace{14mu} 14} \right\rbrack\end{matrix}$

On the other hand, the mutual information between the input symbol U_(i)and the LLR L_(i) for the input symbol may be obtained as follows.

$\begin{matrix}\begin{matrix}{{I\left( {U_{i};\mathcal{L}_{i}} \right)} = {{\mathbb{E}}\left\lbrack {\left. {\log_{2}\left( \frac{2}{1 + e^{- \mathcal{L}_{i}}} \right)} \middle| U_{i} \right. = 0} \right\rbrack}} \\{= {\int_{- \infty}^{\infty}{p\;{\mathcal{L}_{i}\ \left( x \middle| 0 \right)}{\log_{2}\left( \frac{2}{1 + e^{- x}} \right)}{dx}}}} \\{= {1 - {{\mathbb{E}}\left\lbrack {\left. {\log_{2}\left( {1 + e^{- \mathcal{L}_{i}}} \right)} \middle| U_{i} \right. = 0} \right\rbrack}}} \\{= {1 - {\int_{- \infty}^{\infty}{p\;{\mathcal{L}_{i}\left( x \middle| 0 \right)}{\log_{2}\left( {1 + e^{- x}} \right)}{dx}}}}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 15} \right\rbrack\end{matrix}$

Herein, pL_(i)(x) is a probability distribution function of L_(i), andpL_(i)(x|0) is a probability distribution of L_(i) if a given conditionis U_(i)=0. That is, the mutual information may be obtained if only theprobability distribution function of LLR is given. The probabilitydistribution function may be accurately obtained through densityevolution (DE). However, since complexity required for DE is high, itmay be obtained relatively accurately through Gaussian approximation. Inthis case, the mutual information may be expressed as follows. In thefollowing equation, μ_(i) and σ² _(i) corresponding to the input symbolU_(i) may be obtained through Gaussian approximation.

$\begin{matrix}\begin{matrix}{{{I\left( {U_{i};{\mathcal{L}_{i}\left( \mathcal{T}_{M} \right)}} \right)} = {{\mathbb{E}}\left\lbrack {\left. {\log_{2}\left( \frac{2}{1 + e^{- {\mathcal{L}_{i}{(\mathcal{T}_{M})}}}} \right)} \middle| U_{i} \right. = 0} \right\rbrack}},{i \in \mathcal{T}_{M}}} \\{= {\int_{- \infty}^{\infty}{= {\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}e^{- \frac{{({x - \mu_{1}})}^{2}}{2\sigma_{1}^{2}}}{\log_{2}\left( \frac{2}{1 + e^{- x}} \right)}{dx}}}}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 16} \right\rbrack\end{matrix}$1.2 Code Optimization for Maximizing Variance of Mutual Information

Basic concept of a polar code is to polarize mutual information of eachbit channel. Therefore, when a code length is infinite in polar coding,mutual information of a bit channel corresponding to each input bitconverges to 0 or 1, thereby maximizing a variance of mutual informationof all bits.

When a polar code having a length M is constructed based on this basicconcept of the polar code, the variance of the mutual information may bemaximized as follows.

$\begin{matrix}{{{\overset{\sim}{\mathcal{T}}}_{M} = {\arg\;{\max\limits_{\mathcal{T}_{M}}\;{V_{I}\left( \mathcal{T}_{M} \right)}}}},{\mathcal{T}_{M} \in \Psi_{N}^{M}}} & \left\lbrack {{Equation}\mspace{14mu} 17} \right\rbrack\end{matrix}$

Herein, V_(I)(T_(M)) is a variance of mutual information given asfollows.

$\begin{matrix}{{V_{I}\left( \mathcal{T}_{M} \right)} = {{\frac{1}{M}{\sum\limits_{i \in \mathcal{T}_{M}}{I\left( {U_{i};{\mathcal{L}_{i}\left( Y_{\mathcal{T}_{M}} \right)}} \right)}^{2}}} - {M_{I}\left( \mathcal{T}_{M} \right)}^{2}}} & \left\lbrack {{Equation}\mspace{14mu} 18} \right\rbrack\end{matrix}$

Alternatively, it may be optimized as follows by using a variance valueand average value of the mutual information.

$\begin{matrix}{{{\overset{\sim}{\mathcal{T}}}_{M} = {{\arg\;{\max\limits_{\mathcal{T}_{M}}\;{{M_{I}\left( \mathcal{T}_{M} \right)}\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu}{V_{I}\left( \mathcal{T}_{M} \right)}}}} \geq V_{I}^{Th}}},{\mathcal{T}_{M} \in \Psi_{N}^{M}}} & \left\lbrack {{Equation}\mspace{14mu} 19} \right\rbrack\end{matrix}$1.3 Sub-Optimization of Polar Code Having Random Length

A method of calculating mutual information has been described up to nowin regards to two problems related to optimization of a polar code. Asdescribed above, mutual information of LLR may be obtained through DE orGaussian approximation.

Hereinafter, in order to solve the second problem, a method of obtaininga suboptimal solution when a code is designed is proposed in order todecrease complexity for code construction.

First, assume that Δ denotes a step size of the codelength. Thefollowings are several examples of Δ.

$\begin{matrix}{\Delta = 512} & \rightarrow & {N,{M \in \left\{ {512,1024,\ldots}\; \right\}}} \\{\Delta = 256} & \rightarrow & {N,{M \in \left\{ {{256},{512},{768},1024,\ \ldots}\; \right\}}} \\{\Delta = 128} & \rightarrow & {N,{M \in \left\{ {{128},{256},{512},{640},{768},{896},1024,\ldots}\; \right\}}} \\{\Delta = 64} & \rightarrow & {N,{M \in \left\{ {{64},128,\ldots\mspace{14mu},512,576,640,\ldots\mspace{14mu},896,} \right.}} \\\; & \; & \left. {960,1024,\ldots}\; \right\} \\{\Delta = 32} & \rightarrow & {N,{M \in \left\{ {{32},64,{\ldots\mspace{14mu} 512},576,\ldots\mspace{14mu},960,992,} \right.}} \\\; & \; & \left. {1024,\ldots}\; \right\} \\\ldots & \; & \; \\{\Delta = 1} & \rightarrow & {N,{M \in \left\{ {1,2,3,\ldots\mspace{14mu},512\;,513,514,\ldots\mspace{14mu},1022,} \right.}} \\\; & \; & \left. {1023,1024,\mspace{14mu}\ldots} \right\}\end{matrix}$

As shown in the above example, if Δ=1, it is possible to have allavailable code lengths. However, if Δ>1, it may have a code length withan increment corresponding to the value Δ. When polar codes havingvarious lengths are constructed in a real environment, it is sufficientto design only codes having an increment of several constant lengths.Therefore, the value Δ may be determined by considering required severalcode length values when a code is designed in the real environment.

Assume that

${\overset{\sim}{\Psi}}_{\frac{N}{\Delta}}^{\frac{M}{\Delta}}$is a set indicating all cases in which a polar code having a length M isconstructed from a mother polar code having a length N throughcombination of small polar codes having a code length Δ. The number

${\overset{\sim}{\Psi}}_{\frac{N}{\Delta}}^{\frac{M}{\Delta}}$of the sets is as follows.

$\begin{matrix}{{{\overset{\sim}{\Psi}}_{\frac{N}{\Delta}}^{\frac{M}{\Delta}}} = {\begin{pmatrix}\frac{N}{\Delta} \\\frac{M}{\Delta}\end{pmatrix} = \frac{\left( \frac{N}{\Delta} \right)!}{{\left( \frac{M}{\Delta} \right)!}{\left( \frac{N - M}{\Delta} \right)!}}}} & \left\lbrack {{Equation}\mspace{14mu} 19} \right\rbrack\end{matrix}$

When constructing the polar code having the length M, a process offinding a suboptimal solution is as follows.

-   -   Step 1: The greatest common divisor of N and M is found.    -   Step 2: Δ is determined as the found greatest common divisor or        one of values obtained by dividing the greatest common divisor        by a natural number.    -   Step 3: A code for providing maximum mutual information is        selected by finding mutual information for each case in the set

${\overset{\sim}{\Psi}}_{\frac{N}{\Delta}}^{\frac{M}{\Delta}}$in order to construct the polar code having the length M.

The optimization may be expressed by any one of Equations 20 to 22 asfollows.

$\begin{matrix}{{{\overset{\sim}{\mathcal{T}}}_{M}\  = \ {\underset{\mathcal{T}_{M}}{\arg\max}{M_{I}\left( \mathcal{T}_{M} \right)}}},{\mathcal{T}_{M} \in {\overset{\sim}{\Psi}}_{\frac{N}{\Delta}}^{\frac{M}{\Delta}}}} & \left\lbrack {{Equation}\mspace{14mu} 20} \right\rbrack \\{{{\overset{\sim}{\mathcal{T}}}_{M}\  = {\underset{\mathcal{T}_{M}}{\arg\max}{V_{I}\left( \mathcal{T}_{M} \right)}}},{\mathcal{T}_{M} \in {\overset{\sim}{\Psi}}_{\frac{N}{\Delta}}^{\frac{M}{\Delta}}}} & \left\lbrack {{Equation}\mspace{14mu} 21} \right\rbrack \\{{{\overset{\sim}{\mathcal{T}}}_{M} = {{\underset{\mathcal{T}_{M}}{\arg\max}{M_{I}\left( \mathcal{T}_{M} \right)}\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu}{V_{I}\left( \mathcal{T}_{M} \right)}} \geq V_{I}^{Th}}},{\mathcal{T}_{M} \in {\overset{\sim}{\Psi}}_{\frac{N}{\Delta}}^{\frac{M}{\Delta}}}} & \left\lbrack {{Equation}\mspace{14mu} 22} \right\rbrack\end{matrix}$

To facilitate the understanding for the code construction process, twoexamples are described below.

Example 1

When N=8 and M=6

FIG. 2 shows an example of four cases considered when polar codesub-optimization is performed when N=8, M=6, and Δ=2. In addition, FIG.3 shows an example of cases considered when polar code optimization isperformed and not considered when sub-optimization is performed whenN=8, M=6, and Δ=2.

When N=8 and M=6, the greatest common divisor of the two values is 2,and thus Δ=2. In this case, original optimization complexity is asfollows.

$\begin{matrix}{\left| {\overset{\sim}{\Psi}}_{N}^{M} \right| = {\begin{pmatrix}8 \\6\end{pmatrix} = {32}}} & \left\lbrack {{Equation}\mspace{14mu} 23} \right\rbrack\end{matrix}$

Complexity of sub-optimization is as follows.

$\begin{matrix}{\left| {\overset{\sim}{\Psi}}_{\frac{N}{\Delta}}^{\frac{M}{\Delta}} \right| = {\begin{pmatrix}4 \\3\end{pmatrix} = 4}} & \left\lbrack {{Equation}\mspace{14mu} 24} \right\rbrack\end{matrix}$

A mutual information is obtained for each of four cases considered insub-optimization illustrated in FIG. 2. In addition, a case of havingthe maximum mutual information is selected among the obtained mutualinformation. In addition, the same code as in the selected case isconstructed. FIG. 3 shows cases considered when optimization of a polarcode is performed and not considered when sub-optimization is performed.

Example 2

When N=8 and M=4

FIG. 4 shows two cases where mutual information is calculated in polarcode sub-optimization when N=8, M=4, and Δ=4. In addition, FIG. 5 showsan example of six cases where mutual information is calculated in polarcode sub-optimization when N=8, M=4, and Δ=2.

When N=8 and M=6, the greatest common divisor of the two values is 4.Therefore, cases where Δ=2 and Δ=4 are considered. For each case,complexity required for sub-optimization is as follows.

$\begin{matrix}{{{{{When}\mspace{14mu}\Delta} = 4},{\left| {\overset{\sim}{\Psi}}_{\frac{N}{\Delta}}^{\frac{M}{\Delta}} \right| = {\begin{pmatrix}2 \\1\end{pmatrix} = 2}}}{{{{When}\mspace{14mu}\Delta} = 2},{\left| {\overset{\sim}{\Psi}}_{\frac{N}{\Delta}}^{\frac{M}{\Delta}} \right| = {\begin{pmatrix}4 \\2\end{pmatrix} = 6}}}} & \left\lbrack {{Equation}\mspace{14mu} 25} \right\rbrack\end{matrix}$

FIG. 4 and FIG. 5 show cases where mutual information is calculated inpolar code sub-optimization when N=8, M=4, and Δ=4 and when N=8, M=4,and Δ=2.

2. Method of Constructing Polar Code Having Random Length for PhysicalLayer Security

It is described a method by which the aforementioned scheme ofconstructing a polar code having a random length is applied extendedlyto physical layer security communication. An ultimate performanceindicator in the physical layer security is secrecy capacity. Thesecrecy capacity is determined as a difference between channel capacityof a receiver which desires to receive data and an eavesdropper whichdesires not to receive data. Hereinafter, for convenience ofexplanation, a transmitter which transmits data is denoted by Alice, areceiver which desires to receive data is denoted by Bob, and aneavesdropper which desires not to receive data is denoted by Eve.

The secrecy capacity has a meaningful positive (+) value only whenchannel capacity of Bob is greater than channel capacity of Eve. If thechannel capacity of Eve is greater than the channel capacity of Bob, thesecrecy capacity is 0. By using the secrecy capacity as a performanceindicator, the following optimization is performed to construct a polarcode having a length M.

$\begin{matrix}{{{\overset{\sim}{\mathcal{T}}}_{M} = {\arg\;{\max\limits_{\mathcal{T}_{M}}\;{\sum\limits_{i \in \mathcal{T}_{M}}\left( {{I_{b}\left( {U_{i};{\mathcal{L}_{i}\left( \mathcal{T}_{M} \right)}} \right)} - {I_{e}\left( {U_{i};{\mathcal{L}_{i}\left( \mathcal{T}_{M} \right)}} \right)}} \right)^{+}}}}},\mspace{79mu}{\mathcal{T}_{M} \in \Psi_{N}^{M}}} & \left\lbrack {{Equation}\mspace{14mu} 26} \right\rbrack\end{matrix}$

Herein, I_(b)(·) denotes channel capacity (or mutual information) ofBob, and I_(e)(·) denotes channel capacity (or mutual information) ofEve. In addition, the function (x)⁺ is defined as follows.

$\begin{matrix}{(x)^{+} = \left\{ \begin{matrix}{x,} & {{{if}\ x} > 0} \\{0,} & {otherwise}\end{matrix} \right.} & \left\lbrack {{Equation}\mspace{14mu} 27} \right\rbrack\end{matrix}$

The mutual information for Bob and Eve may be calculated by a methodusing the aforementioned LLR.

If the length of the polar code is infinitely long and thus channelpolarization is completely achieved, a security polar code may beeffectively constructed through optimization according to Equation 26.However, if the length of the polar code is finite and thus the channelpolarization is not completely achieved, channel capacity calculatedthrough optimization according to Equation 26 may not have a meaning ofa transfer rate for transmitting data while maintaining securitysubstantially without error. In order to transmit data to Bob withouterror, mutual information of Bob must be very close to 1, and in orderto transmit data to Bob while maintaining security, mutual informationof Eve must be very close to 0.

The following descriptions are several examples for a possibility ofmaintaining security on the basis of the mutual information of Bob andEve.

Example 3.1

I_(b) (I_(i); L_(i)(T_(M)))=0.6, I_(e) (U_(i); L_(i)(T_(M)))=0.3

In this case, data decoded by Bob may have an error. In addition,security for Eve is not maintained. That is, Eve may also decode a partof data.

Example 3.2

I_(b) (U_(i); L_(i)(T_(M)))=0.9999, I_(e)(U_(i); L_(i)(T_(M)))=0.3

In this case, data decoded by Bob has almost no error. However, securityfor Eve is not maintained.

Example 3.3

I_(b) (U_(i); L_(i)(T_(M)))=0.6, I_(e) (U_(i); L_(i)(T_(M)))=0.0001

In this case, data decoded by Bob may have an error. However, securityfor Eve is maintained.

Example 3.4

I_(b) (U_(i); L_(i)(T_(M)))=0.9999, I_(e) (U_(i); L_(i)(T_(M)))=0.0001

In this case, data decoded by Bob has almost no error. In addition,security for Eve is maintained.

Considering the aforementioned examples 3.1 to 3.4, the secrecy capacityaccording to Equation 26 is calculated as follows.Σ(I _(b)(U _(i);

_(i)(

_(M)))−I _(B)(U _(B);

_(i)(

_(M)))=(0.6−0.3)+(0.9999−0.3)+(0.6−0.0001)+(0.9999−0.0001)  [Equation28]

However, among the aforementioned four examples, the examples 3.1 to 3.3are cases where data transmission has an error or security is notmaintained. When the secrecy capacity is determined in practice, theexamples 3.1 to 3.3 must be excluded. Therefore, Equation 26 may bemodified as follows.

$\begin{matrix}{{{\overset{\sim}{\mathcal{T}}}_{M} = {\arg{\max\limits_{\mathcal{T}_{M}}{\sum\limits_{i \in \mathcal{T}_{M}}\left( {{I_{b}\left( {U_{i};{\mathcal{L}_{i}\left( \mathcal{T}_{M} \right)}} \right)} - {I_{e}\left( {U_{i};{\mathcal{L}_{i}\left( \mathcal{T}_{M} \right)}} \right)}} \right)^{+ \mu}}}}},\mspace{79mu}{\mathcal{T}_{M} \in \Psi_{N}^{M}}} & \left\lbrack {{Equation}\mspace{14mu} 29} \right\rbrack\end{matrix}$

Herein, the function (x)⁺ is as follows.

$\begin{matrix}{(x)^{+ \mu} = \left\{ \begin{matrix}{x,\ {{{if}\ x} \geq \mu}} \\{0,\ {otherwise}}\end{matrix} \right.} & \left\lbrack {{Equation}\mspace{14mu} 30} \right\rbrack\end{matrix}$

Considering only a case where data transmission has no error andsecurity is maintained, μ must be set to a value close to 1 in order todetermine the secrecy capacity. For example, it may be set to μ=0.9999.The following is an example of calculating the secrecy capacity in theaforementioned example 3.1, by considering μ=0.9998.

$\begin{matrix}{\sum\left( {{{I_{b}\left( {U_{i};{\mathcal{L}_{i}\left( \mathcal{T}_{M} \right)}} \right)} - {I_{e}\left( {U_{i};{\mathcal{L}_{i}\left( \mathcal{T}_{M} \right)}} \right)}^{+ \mu}} = {{\left( {0.6 - {0.3}} \right)^{+ \mu} + \left( {{{0.9}999} - {0.3}} \right)^{+ \mu} + \left( {0.6 - {{0.0}001}} \right)^{+ \mu} + \left( {{{0.9}999} - {{0.0}001}} \right)^{+ \mu}} = \left( {0.9999 - {{0.0}001}} \right)}} \right.} & \left\lbrack {{Equation}\mspace{14mu} 31} \right\rbrack\end{matrix}$2.1 Sub-Optimization of Polar Code Having Random Length for PhysicalLayer Security

A sub-optimization method for decreasing complexity required when apolar code having a random length without consideration of physicallayer security may also equally apply to a case where a security polarcode is constructed by considering the physical layer security.

In this case, in order to determine optimal T_(M), optimization isperformed by considering only cases of being included in a set

${\overset{\sim}{\Psi}}_{\frac{N}{\Delta}}^{\frac{M}{\Delta}},$instead of considering all cases of being included in a set Ψ_(N) ^(M).The optimization may be expressed by any one of the following Equations32 and 33.

$\begin{matrix}{{{\overset{\sim}{\mathcal{T}}}_{M} = {\arg{\max\limits_{\mathcal{T}_{M}}{\sum\limits_{i \in \mathcal{T}_{M}}\left( {{I_{b}\left( {U_{i};{\mathcal{L}_{i}\left( T_{M} \right)}} \right)} - {I_{e}\left( {U_{i};{\mathcal{L}_{i}\left( \mathcal{T}_{M} \right)}} \right)}} \right)^{+}}}}},\mspace{79mu}{\mathcal{T}_{M} \in {\overset{\sim}{\Psi}}_{\frac{N}{\Delta}}^{\frac{M}{\Delta}}}} & \left\lbrack {{Equation}\mspace{14mu} 32} \right\rbrack \\{{{\overset{\sim}{\mathcal{T}}}_{M} = {\arg{\max\limits_{\mathcal{T}_{M}}{\sum\limits_{i \in \mathcal{T}_{M}}\left( {{I_{b}\left( {U_{i};{\mathcal{L}_{i}\left( T_{M} \right)}} \right)} - {I_{e}\left( {U_{i};{\mathcal{L}_{i}\left( \mathcal{T}_{M} \right)}} \right)}} \right)^{+ \mu}}}}},\mspace{79mu}{\mathcal{T}_{M} \in {\overset{\sim}{\Psi}}_{\frac{N}{\Delta}}^{\frac{M}{\Delta}}}} & \left\lbrack {{Equation}\mspace{14mu} 33} \right\rbrack\end{matrix}$3. HARQ Based on Polar Coding

A new HARQ scheme based on polar coding as described above will bedescribed. First, it is defined that N denotes a length of a mothercode, and K denotes the number of information bits included in themother code. In addition, I denotes a set of information bits in themother code, and Z denotes a set of frozen bits. In general, the frozenbit may use 0 (zero). N coded bits of the given mother code are dividedinto J sets, each set is represented by

_(j), j=1, 2, . . . , J, and a size of each set is represented by M_(j)as follows.|

₁ |=M ₁,|

₂ |=M ₂, . . . ,|

_(J) |=M _(J)

Herein, Σ_(j=1) ^(J) M_(j)=N is satisfied, and

₁, ∩

₂=θ, l₁≠l₂ is satisfied.

The transmitter transmits only coded bits belonging to

₁ in initial transmission. Upon receiving NACK (or upon not receivingACK) from the receiver, the transmitter transmits coded bits belongingto

₂. Similarly, upon receiving NACK after transmitting coded bitsbelonging to

_(j), the transmitter transmits

_(j+1). In addition, if the transmitter receives ACK from the receiver,a new mother code is constructed to transmit next new information bits,and new HARQ is performed again.

3.1 HARQ Scheme 1

One way to construct an optimized HARQ scheme is to optimize

_(j) such that mutual information is maximized as follows.

$\begin{matrix}\begin{matrix}\mathcal{G}_{1}^{opt} & = & {\arg{\max\limits_{\mathcal{G} \subseteq {\lbrack{1:N}\rbrack}}{I\left( {U_{\mathcal{I}};Y_{\mathcal{G}_{1}}} \right)}}} \\\mathcal{G}_{2}^{opt} & = & {\arg{\max\limits_{\mathcal{G}_{2} \subseteq {{\lbrack{1:N}\rbrack}\backslash\mathcal{G}_{1}^{opt}}}{I\left( {U_{\mathcal{I}};Y_{\mathcal{G}_{2}\bigcup\mathcal{G}_{1}^{opt}}} \right)}}} \\G_{3}^{opt} & = & {\arg\;{\max\limits_{\mathcal{G}_{3} \subseteq {{\lbrack{1:N}\rbrack}{{\backslash(}{{\mathcal{G}_{1}^{opt}\bigcup\mathcal{G}_{2}^{opt}})}}}}{I\left( {U_{\mathcal{I}};Y_{\mathcal{G}_{3}\bigcup\mathcal{G}_{1}^{opt}\bigcup\mathcal{G}_{2}^{opt}}} \right)}}} \\\; & \vdots & \; \\G_{J}^{opt} & = & {\arg\;{\max\limits_{\mathcal{G}_{J} \subseteq {{\lbrack{1:N}\rbrack}{{\backslash(}{{\bigcup_{j = 1}^{J - 1}\mathcal{G}_{j}^{opt}})}}}}{I\left( {U_{\mathcal{I}};Y_{\mathcal{G}_{J}\bigcup{\bigcup_{j = 1}^{J - 1}\mathcal{G}_{j}^{opt}}}} \right)}}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 35} \right\rbrack\end{matrix}$

Herein, mutual information I(U_(I); Y_(B)) is calculated as follows.

$\begin{matrix}{{I\left( {U_{\mathcal{I}};Y_{\mathcal{B}}} \right)} = {\sum\limits_{i \in \mathcal{I}}{I\left( {U_{i};{\mathcal{L}_{i}(\mathcal{B})}} \right)}}} & \left\lbrack {{Equation}\mspace{14mu} 36} \right\rbrack\end{matrix}$

In particular, as described above, I(U_(i); L_(i)(B)) is calculated asfollows.

$\begin{matrix}\begin{matrix}{{I\left( {U_{i};{\mathcal{L}_{i}(\mathcal{B})}} \right)} = {{\mathbb{E}}\left\lbrack {I_{h_{b}}\left( {U_{i};{\mathcal{L}_{i}(\mathcal{B})}} \right)} \right\rbrack}} \\{{= {{\mathbb{E}}\left\lbrack {\left. {\log_{2}\left( \frac{2}{1 + e^{- {\mathcal{L}_{i}{(\mathcal{B})}}}} \right)} \middle| U_{i} \right. = 0} \right\rbrack}},} \\{i \in \mathcal{B}} \\{= {\int_{- \infty}^{\infty}{{p\left( \mu_{i} \right)}{\int_{- \infty}^{\infty}{\frac{1}{\sqrt{2{\pi\sigma}_{i}^{2}}}e^{- \frac{{({x - \mu_{i}})}^{2}}{2\sigma_{i}^{2}}}}}}}} \\{{\log_{2}\left( \frac{2}{1 + e^{- x}} \right)}d\; x\; d\;\mu_{i}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 37} \right\rbrack\end{matrix}$

In order to determine the set

₁ for the initial transmission, the following properties may bemathematically proved.

$\begin{matrix}{{I\left( {U_{\mathcal{I};}Y_{\mathcal{I}}} \right)} = {\underset{{\mathcal{G} \in {\lbrack{1:N}\rbrack}},\;{{\mathcal{G}} = K}}{\max\;}{I\left( {U_{\mathcal{I};}Y_{\mathcal{G}}} \right)}}} & \left\lbrack {{Equation}\mspace{14mu} 38} \right\rbrack\end{matrix}$

Therefore,

₁ is allowed to include all information bits in a set I as follows.

₁⊇

  [Equation 39]

A calculation amount required to optimize all sets

_(j) in this manner is as follows.

$\begin{matrix}{\begin{pmatrix}N \\M_{1}\end{pmatrix} + \begin{pmatrix}{N - M_{1}} \\M_{2}\end{pmatrix} + \begin{pmatrix}{N - M_{1} - M_{2}} \\M_{3}\end{pmatrix} + \ldots + \begin{pmatrix}M_{J} \\M_{J}\end{pmatrix}} & \left\lbrack {{Equation}\mspace{14mu} 40} \right\rbrack\end{matrix}$

In order to decrease the calculation amount, a method proposed above insub-optimization of a polar code having a random length may be used.That is, a minimum increment ΔN of a code length may be defined, and acode may be optimized such that ΔN corresponds to a resolution. In thiscase, a total calculation amount is as follows.

$\begin{matrix}{\begin{pmatrix}\frac{N}{\Delta N} \\\frac{M_{1}}{\Delta N}\end{pmatrix} + \begin{pmatrix}\frac{N - M_{1}}{\Delta N} \\\frac{M_{2}}{\Delta\; N}\end{pmatrix} + \begin{pmatrix}\frac{N - M_{1} - M_{2}}{\Delta\; N} \\\frac{M_{3}}{\Delta N}\end{pmatrix} + \ldots + \begin{pmatrix}\frac{M_{J}}{\Delta N} \\\frac{M_{J}}{\Delta N}\end{pmatrix}} & \left\lbrack {{Equation}\mspace{14mu} 41} \right\rbrack\end{matrix}$3.2 Simplified HARQ Scheme 1

Now, a simplified HARQ scheme based on grouping is described. Such ascheme is simpler than the aforementioned HARQ scheme 1.

In polar coding, mutual information I(W_(N) ^((i))) of an i^(th) bitchannel W_(N) ^((i)) is as follows.I(W _(N) ^((i)))=I(U _(i) ;Y ₁ ^(N) ,U ₁ ^(i−1)),i=1, . . .,N  [Equation 42]

To decrease a calculation amount of puncturing, a method is used inwhich bits are selected sequentially one by one in order of significanceamong N bit channels. That is, a method is used in which one bit channeloutput Y_(j) is selected as follows, in order of significance amongoutputs Y₁ ^(N) of all N bit channels.Y ₁ ^(N) →Y _(j)  [Equation 43]

To decode an i^(th) input bit U_(i), polar coding uses all output bitsY₁ ^(N) and previously decoded all input bits U₁ ^(i−1). However, theremay be errors in input bits which are previously decoded in a realenvironment. To solve such a problem, a new bit grouping algorithm isproposed as follows.

(Step 1)

It is assumed that indices i₁, i₂, . . . , i_(N) express an index of abit channel when the bit channel is aligned in such a manner that mutualinformation is decreased as follows.I(U _(i) _(N) ;Y _(i) ₁ ^(i) ^(N) ,U _(i) ₁ ^(i) ^(N−1) )≥I(U _(i)_(N−1) ;Y _(i) ₁ ^(i) ^(N) ,U _(i) ₁ ^(i) ^(N−2) )≥ . . . ≥I(U _(i) ₂ ;Y_(i) ₁ ^(i) ^(N) ,U _(i) ₁ )≥I(U _(i) ₁ ;Y _(i) ₁ ^(i) ^(N) )  [Equation44]

The Equation 44 above is simply expressed as follows.I(W _(N) ^((i) ^(N) ⁾)≥I(W _(N) ^((i) ^(N−1) ⁾)≥ . . . I(W _(N) ^((i) ²⁾)≥I(W _(N) ^((i) ¹ ⁾)  [Equation 45](Step 2)

The most significant input bits are sequentially selected one by one onthe basis of mutual information, and the selected input bit and oneoutput bit having the greatest mutual information are determined. Such aselection method may be mathematically obtained as follows.

-   -   U_(i) _(N) is an input bit which generates the greatest mutual        information I (W_(N) ^((i) ^(j) ⁾) among all input bits. When an        output bit having the greatest mutual information with respect        to the input bit U_(i) _(N) is expressed by Y _(jN), an index jN        of an output bit Y _(jN) is as follows.

$\begin{matrix}{{\overset{\sim}{j}}_{N} = {\arg{\max\limits_{j \in {\{{1,2,\;\ldots\;,\; N}\}}}{I\left( {{U_{i_{N}};Y_{j}},{\overset{\hat{}}{U}}_{i_{1}}^{i_{N - 1}}} \right)}}}} & \left\lbrack {{Equation}\mspace{14mu} 46} \right\rbrack\end{matrix}$

Herein, Û_(i) ₁ ^(i) ^(N−1) denotes decoded values of previous inputbits U_(i) ₁ ^(i) ^(N−1) when an output Y_(j) is given.

-   -   Next, is an input bit which generates the second greatest mutual        information. When an output bit having the greatest mutual        information with respect to the input bit U_(i) _(N−1) is        expressed by Y _(jN−1) , an index j _(N−1) of an output bit Y        _(jN−1) is as follows.

$\begin{matrix}{{\overset{\sim}{j}}_{N - 1} = {\arg{\max\limits_{j \in {{\{{1,2,\;\ldots\;,\; N}\}}/{\overset{\sim}{j}}_{N}}}{I\left( {{U_{i_{N - 1}};Y_{j}},{Y_{{\overset{\sim}{j}}_{N}}{\overset{\hat{}}{,U}}_{i_{1}}^{i_{N - 2}}}} \right)}}}} & \left\lbrack {{Equation}\mspace{14mu} 47} \right\rbrack\end{matrix}$

Herein, Û_(i) ₁ ^(i) ^(N−2) denotes decoded values of previous inputbits U_(i) ₁ ^(i) ^(N−2) when an output Y_(j) and an output Y _(jN)optimally selected previously are given.

-   -   In the aforementioned manner, the most significant input bits        are selected sequentially one by one on the basis of mutual        information, and the selected input bit and one output bit        having the greatest mutual information are determined. This is        generalized by the following equation.

$\begin{matrix}{{{\overset{\sim}{j}}_{k} = {\arg{\max\limits_{j \in {{\{{1,2,\;\ldots\;,\; N}\}}/{\{{{\overset{\sim}{j}}_{N},\;\ldots\;,\;{\overset{\sim}{j}}_{k + 1}}\}}}}{I\left( {{U_{i_{k}};Y_{j}},Y_{{\overset{\sim}{j}}_{k + 1}}^{{\overset{\sim}{j}}_{N}},{\overset{\hat{}}{U}}_{i_{1}}^{i_{k - 1}}} \right)}}}},\mspace{79mu}{k = 1},2,\ldots\mspace{14mu},N} & \left\lbrack {{Equation}\mspace{14mu} 48} \right\rbrack\end{matrix}$

Herein, Û_(i) ₁ ^(i) ^(k−1) denotes decoded values of previous inputbits U_(i) ₁ ^(i) ^(k−1) when an output Y_(j) and an output Y _(j)_(k+1) ^(jN) optimally selected previously are given.

Therefore, a selection scheme for optimization may be mathematicallyobtained, and a result thereof is as follows.j _(k) =i _(k) ,k=1,3, . . . ,N  [Equation 49]

Such a result means that an output bit which generates the greatestmutual information with respect to each input bit has the same index. Inthis case, it is assumed that a matrix of generating a polar code isgiven by. Herein, it is a 2×2 matrix given by F₂=[1 1;1−1]. By usingsuch a mathematical result, optimal index sets may be mathematicallyexpressed as follows.

[Simplified Bit Grouping Algorithm for HARQ Scheme 1]

FIG. 6 shows an example of a simplified bit grouping algorithm for anHARQ scheme 1. In addition, FIG. 7 is a conceptual diagram forexplaining concept of a simplified bit grouping algorithm for an HARQscheme 1.

$\begin{matrix}\begin{matrix}\mathcal{G}_{1} & = & {\underset{\underset{{for}\mspace{11mu}{({M_{1} - K})}\mspace{11mu}{zeros}}{︸}}{\left\lbrack {i_{N - M_{1} + 1},i_{N - M_{1} + 2},\ldots\mspace{14mu},i_{N - K}} \right.},} \\\; & \; & \underset{\underset{{for}\mspace{11mu} K\mspace{11mu}{info}\mspace{11mu}{bits}}{︸}}{\left. {i_{N - K + 1},i_{N - K + 2},\ldots\mspace{14mu},i_{N}} \right\rbrack} \\\mathcal{G}_{2} & = & \underset{\underset{{for}\mspace{11mu} M_{2}\mspace{11mu}{zeros}}{︸}}{\left\lbrack {i_{N - M_{1} - M_{2} + 1},i_{N - M_{1} - M_{2} + 2},\ldots\mspace{14mu},i_{N - M_{1}}} \right\rbrack} \\\mathcal{G}_{3} & = & \underset{\underset{{for}\mspace{11mu} M_{3}\mspace{11mu}{zeros}}{︸}}{\left\lbrack {i_{N - M_{1} - M_{2} - M_{3} + 1},i_{N - M_{1} - M_{2} - M_{3} + 2},\ldots\mspace{14mu},i_{N - M_{1} - M_{2}}} \right\rbrack} \\\; & \vdots & \; \\\mathcal{G}_{J} & = & \underset{\underset{{for}\mspace{11mu} M_{J}\mspace{11mu}{zeros}}{︸}}{\left\lbrack {i_{1},i_{2},\ldots\mspace{14mu},i_{N - {\sum\limits_{j = 1}^{J}M_{j}}}} \right\rbrack}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 50} \right\rbrack\end{matrix}$

The simplified bit grouping algorithm and its concept for the HARQscheme 1 according to Equation 50 are as shown in FIG. 6 and FIG. 7.

Although the locations of the information bit and the frozen bit aredetermined based on the aforementioned mutual information or ranking ofthe mutual information in the HARQ schemes proposed in the presentspecification, it is apparent that the HARQ scheme are also applicableto a case where the locations of the information bit and the frozen bitare determined based on any other criteria.

3.3 HARQ Scheme 2

In the HARQ scheme 1 described above, the first step is to construct anoptimal mother code having a length N and having K information bits.Therefore, a location of the information bit is optimized based on afirst constructed mother code. However, the location of the informationbit is not optimized in terms of a first packet

₁. More generally, the location of the information bit may be optimizedonly when all of J packets

_(j), j=1, 2, . . . , J are transmitted.

To solve this problem, the HARQ scheme 2 considers an approach oppositeto the HARQ scheme 1. In the HARQ scheme 2, the information bit isoptimized based on a first packet (or a coded bit group). Iftransmission of the first packet is not successful, additional packetsare transmitted. The HARQ scheme 2 may be classified into three schemesaccording to a scheme of transmitting an additional packet.

3.3.1 HARQ Scheme 2A

In the HARQ scheme 2A, locations of information bits of a polar code areoptimized based on a first packet. That is, in the HARQ scheme 2A,packets in which a location of an information bit having a length M₁ isoptimized are first transmitted, and all packets transmittedadditionally thereafter are frozen bits (or zeros). In this case, codedbits of the additionally transmitted packet are not independent of codedbits of a previously transmitted packet, and coded packets of allpackets constitute one large polar code.

FIG. 8 shows an example of transmission based on the HARQ scheme 2A.

As shown in FIG. 8, a location of an information bit in a first packetis optimized based on the first packet given in a current state. Afterthe first packet is transmitted, if a receiver fails to properly decodethe packet, a transmitting end additionally transmits a second packet.In this case, input bits of the second packet are all frozen bits, butoutput bits of the second bit depend on input bits of the first packet.That is, as shown in FIG. 8, one large polar code (having a length of 16in FIG. 8) is constructed in which the first packet and the secondpacket are combined. In addition, decoding is performed again in areceiving end by using all received bits.

3.3.2 HARQ Scheme 2B

In the HARQ scheme 2B, similarly to the HARQ method 2A, locations ofinformation bits of a polar code are optimized based on a first packet.However, the HARQ scheme 2B differs from the HARQ scheme 2A in that anadditionally transmitted packet transmits specific information bitstogether in addition to a frozen bit. The information bits transmittedthrough the additionally transmitted packet are information bitstransmitted through a bit channel having low mutual information amonginformation bits transmitted through the existing packet. That is, theinformation bits transmitted through the additionally transmitted packetare a type of repetition coding.

FIG. 9 shows an example of transmission based on the HARQ scheme 2B.

As shown in FIG. 9, a location of an information bit of a first packetis the same as that of the HARQ scheme 2A. When a second packet isconstructed after transmitting the first packet, one information bit(data 5) is transmitted together with a frozen bit in the HARQ scheme2B.

The reason for retransmitting the data 5 among 5 information bitstransmitted through the first packet is as follows. First, five inputinformation bits in the first packet are transmitted on bit channelseach having a different mutual information. Data 1 is transmittedthrough a bit channel having mutual information of 0.9961. Data 2 istransmitted through a bit channel having mutual information of 0.8789.Data 3 is transmitted through a bit channel having mutual information of0.8086. Data 4 is transmitted through a bit channel having mutualinformation of 0.6836. Data 5 is transmitted through mutual informationof 0.3164. When a polar code having a length of 16 is constructed totransmit the second packet, mutual information of bit channels on which5 information bits are transmitted is improved. The mutual informationis improved to each of bit channels in which the mutual information ofdata 1 is 1.0000, the mutual information of data 2 is 0.9922, the mutualinformation of data 3 is 0.9853, the mutual information of data 4 is0.9634, and the mutual information of data 5 is 0.7725. That is, in allfive bit channels, the mutual information is more improved than a casewhere only the first packet is transmitted alone.

On the basis of mutual information of a second packet portion, mutualinformation of 8 bit channels in a packet is small in general incomparison with the first packet. However, there may be exceptions. Forexample, as shown in FIG. 9, mutual information of an 8^(th) bit channelof the second packet is 0.8999, which is greater than mutual informationof 0.7725 for a bit channel on which data 5 is transmitted in the firstpacket. Basic concept of polar coding is to transmit information bitsthrough bit channels with highest mutual information.

Therefore, a frozen bit of the HARQ scheme 2A is transmitted through the8^(th) bit channel of the second packet, which is not effective. Inorder to solve such a problem, in the HARQ scheme 2B, data 5 transmittedon a bit channel having lower mutual information among information bitstransmitted in the first packet is retransmitted through the 8^(th) bitchannel of the second packet. That is, in comparison with the mutualinformation of bit channels for transmitting information bits in thefirst packet, if there is a bit channel having mutual information with ahigher priority in the second packet, the information bit may berepeatedly transmitted through the bit channel

3.3.3 HARQ Scheme 2C

The HARQ scheme 2C is the same as the HARQ scheme 2A and the HARQ scheme2B in a sense that locations of information bits of a polar code areoptimized based on a first packet. However, the HARQ scheme 2C differsfrom the HARQ scheme 2B in that new information bits are transmittedthrough a bit channel with high mutual information among bit channelsbelonging to an additionally transmitted packet.

FIG. 10 shows an example of transmission according to the HARQ scheme2C.

As shown in FIG. 10, mutual information of an 8^(th) bit channel of asecond packet is higher than mutual information of a bit channel onwhich data 5 of a first packet is transmitted. Therefore, it is moreefficient to transmit an information bit than to transmit a frozen bitthrough the 8-bit channel of the second packet. In the HARQ scheme 2B, aprevious information bit is retransmitted through the 8^(th) bit channelof the second packet. However, in the HARQ scheme 2C, a new informationbit, i.e., data 6, is transmitted through the 8^(th) bit channel of thesecond packet. That is, in comparison with the mutual information of bitchannels for transmitting information bits in the first packet, if thereis a bit channel having mutual information with a higher priority in thesecond packet, the information bit may be repeatedly transmitted throughthe bit channel.

FIG. 11 shows an example of comparing the HARQ scheme 2A, the HARQscheme 2B, and the HARQ scheme 2C.

As shown in FIG. 11, an 8^(th) bit of a second packet is used totransmit a frozen bit in the HARQ scheme 2A, to transmit a previousinformation bit in the HARQ scheme 2B, and to transmit a new informationbit in the HARQ scheme 2C.

3.3.4 Combination of the HARQ Scheme 1 and the HARQ Scheme 2

A scheme may be considered in which the HARQ scheme 1 and the HARQscheme 2 are combined complexly. First, advantages and disadvantages ofthe HARQ scheme 1 and the HARQ scheme 2 are compared.

Since an information bit is optimized based on a mother code in the HARQscheme 1, a location of the information bit may be significantlydifferent from an optimal location when only an initial packet istransmitted. Therefore, although a significantly large mother code isconstructed and thus many packets are generated, performancedeterioration may occur when only the initial packets are transmitted.

On the other hand, an information bit is optimized based on a firstpacket in the HARQ scheme 2. However, when many packet are additionallytransmitted after the first packet, locations of the information bitsmay be significantly different from an optimal location, which may leadto performance deterioration.

FIG. 12 shows an example of a scheme in which the HARQ scheme 1 and theHARQ scheme 2 are combined.

As shown in FIG. 12, the scheme in which the HARQ scheme 1 and the HARQscheme 2 are combined are initially performed based on the HARQ scheme1, but is performed based on the HARQ scheme 2 after all coded bits of amother code are transmitted. In this case, an information bit isoptimally configured neither when a first packet is transmitted norafter all packets are transmitted. That is, in the HARQ scheme 1, theinformation bit is optimally configured when a last packet istransmitted, and in the HARQ scheme 2, the information bit is optimallyconfigured when a first packet is transmitted. However, the scheme inwhich the HARQ scheme 1 and the HARQ scheme 2 are combined may variablycoordinate a location of a packet in which the information bit isoptimally configured. The optimal packet location may be determined byconsidering several conditions such as a channel environment or thelike.

When it is said that the HARQ scheme 1 is used first, followed byswitching to the HARQ scheme 2, it is a description in terms of agenerator matrix of a polar code, not in terms of a location of aninformation bit. When switching from the HARQ scheme 1 to the HARQscheme 2, the location of the information bit is not changed.

FIG. 13 shows another example of a scheme in which the HARQ scheme 1 andthe HARQ scheme are combined.

As shown in FIG. 13, an operation is first performed based on the HARQscheme 1. Specifically, a mother code having a length of 8 is generatedbased on the HARQ scheme 1, and a location of an information bit isconfigured in the generated mother code in an optimal manner. Last 5bits are configured in the example of FIG. 13. The mother code in whichthe location of the information bit is configured is punctured togenerate a first frame having a length of 6 and a second frame having alength of 2. In addition, two generated frames (or packets) aresequentially transmitted. When all of the two frames are transmitted, anoperation is performed based on the HARQ scheme 2. As such, whenswitching to the HARQ scheme 2, the location of the information bit isnot changed. Since the information bit has already been transmittedpreviously, it cannot be changed and should not be changed. As shown inFIG. 13, last 5 bits (last 4 bits and a 9^(th) bit from the end, on thebasis of a polar code having a length of 16) is still the informationbit. After switching to the HARQ scheme 2, frozen bits are transmittedin a state where a location of the existing information bit is fixed.

3.3.5 Improved HARQ Scheme 2

Another problem of the HARQ scheme 2 is that a generator matrix of apolar code may be different from an optimal polar code generator matrixover time. Herein, the generator matrix of the optimal polar code existswhen a code length is 2^(n), and may imply a generator matrix of anoriginal polar code. More specifically, in the HARQ scheme 2, the morethe packets are transmitted, the longer the polar code are constructedin a receiving end. In this case, a polar code to be constructed latermust include a generator matrix of a previously constructed polar code.Herein, if no condition is given, a generator matrix of a polar codeconstructed later may be different from a generator matrix of an optimalpolar code. In order to solve such a problem, packets are additionallytransmitted based on HARQ so that a polar code to be constructed is anoptimal polar code whenever a length of a polar code to be constructedin a receiving end is 2^(n), n=1, 2, 3, . . . .

Herein, optimization implies not optimization of the location of theinformation bit but optimization of the generator matrix of the polarcode. For example, the HARQ scheme 2 for the optimal generator matrixmay be configured as follows. First, an optimal generator matrixcorresponding to a length 2³=8 and an optimal generator matrixcorresponding to a length 2⁴=16 are configured. Thereafter, a generatormatrix having a length M₁ and a length M₂ are constructed. Herein,8<M₁<M₂<16.

In this case, when constructing the generator matrix having the lengthM₁ and the length M₂, the following conditions shall be satisfied.

1) The generator matrix having the length M₁ shall include a generatormatrix having a length of 8, and the generator matrix having the lengthM₁ shall be included in the generator matrix having the length M₂.

2) The generator matrix having the length M₂ shall include the generatormatrix having the length M₁, and shall be included in the generatormatrix of the length of 16.

Transmission is achieved in practice as follows.

First, a polar code having a length of 8 is transmitted. Next, in thepolar code having the length M₁, the remaining parts except for agenerator matrix of a part corresponding to a previously transmittedpolar code of a length of 8 are transmitted. Next, in the polar codehaving the length M₂, the remaining parts except for a generator matrixof a part corresponding to the previously transmitted polar code of thelength M₁ are transmitted. Finally, in the polar code having a length of16, the remaining parts except for a generator matrix of a partcorresponding to the polar code having the length M₂ are transmitted.

3.4 HARQ Scheme 3

A new HARQ scheme different from the aforementioned HARQ scheme 1 andHARQ scheme 2 is considered. The HARQ scheme 3 is the same as the HARQscheme 1 in a sense that a mother code is generated and the generatedmother code is punctured. However, in the HARQ scheme 1, the mother codeis generated, and puncturing is performed after locations of aninformation bit and a frozen bit are optimized. On the other hand, inthe HARQ scheme 3, the locations of the information bit and the frozenbit are optimized after the mother code is generated and puncturing isperformed.

For example, after a mother code with a transfer rate of ⅓ isconstructed, if it is assumed that a reference transfer rate for HARQtransmission is ½, the HARQ scheme 3 preferentially generates a codewith a transfer rate of ½ by puncturing the mother code, and thereafteroptimizes the locations of the information bit and the frozen bit. Inthis case, the reference transfer rate (e.g., ½ or ⅓) for HARQtransmission may be configured by a previously received indicationsignal, or may be predetermined.

According to the aforementioned HARQ scheme 3, the location of theinformation bit and the location of the frozen bit are not optimized forevery puncturing, but are optimized by assuming a case where a mothercode is punctured based on a specific pattern with a coding rate. Bydirectly applying the determined locations of the information bit andfrozen bit, transmission is achieved in such a manner that, for eachHARQ packet transmission, a part of the mother code is punctured or themother code is repeated.

4. Secure HARQ Based on Polar Coding

A scheme is described in which the aforementioned polar coding basedHARQ schemes is extendedly applied to physical layer securitycommunication. Data transmitted by Alice (i.e., a transmitter) must bedecoded without error by Bob (i.e., a receiver), and must not be decodedby Eve (i.e., an eavesdropper). For this, mutual information of achannel between Alice and Bob must be kept very close to 1, and mutualinformation of a channel between Alice and Eve must be kept very closeto 0.

The following description assumes that channel capacity for Bob andchannel capacity for Eve are not known to Alice. If the channel capacityfor Bob and the channel capacity for Eve are known to Alice, a polarcode may be constructed based on the existing method and there is noneed to use a secure HARQ scheme. Instead, it is assumed that Aliceknows that the channel capacity for Bob is at least greater than C^(b)_(min), and Alice knows that the channel capacity for Eve is at bestless than C^(c) _(max). In a real environment, the values C^(b) _(min)and C^(c) _(max) may be conservatively estimated.

A polar code of which a code length is N and the number of informationbits is K is constructed as a mother code. It is assumed that Z denotesa set of frozen bits, I denotes a set of information bits, and R denotesa set of random bits. In addition, it is assumed that bit channelindices i₁, i₂, . . . , i_(N) respectively denote indices of bitchannels which are sorted in a descending order of mutual information ofa bit channel.I(W _(N) ^(i) ^(N) ⁾)≥I(W _(N) ^((i) ^(N−1) ⁾)≥ . . . I(W _(N) ^((i) ²⁾)≥I(W _(N) ^((i) ¹ ⁾)  [Equation 50]

In this case, sets I, Z, and R are as follows.

={i _(k) :i ₁ ≤i _(k) ≤i _({circumflex over (b)}) }={i ₁ ,i ₂ , . . . ,i_({circumflex over (b)})}

={i _(k) :i _({circumflex over (b)}) <i _(k) <i _(ê) }={i_({circumflex over (b)}+1) ,i _({circumflex over (b)}+2) , . . . ,i_(ê−1)}

={i _(k) :i _(ê) ≤i _(k) ≤i _(N) }={i _(ê) ,i _(ê+1) , . . . ,i_(N)}  [Equation 51]

Herein, i_({circumflex over (b)}) representing a boundary between Z andI and i_(ê) representing a boundary between I and R may be determined asfollows.

$\begin{matrix}{{i_{\hat{b}} = {\max\limits_{i \in {\lbrack{i_{1}:i_{N}}\rbrack}}{i\mspace{14mu}{subject}\mspace{14mu}{to}}}}\mspace{14mu}{{I\left( {U_{i};\left. {\mathcal{L}_{i}^{b}\left( \left\lbrack {1:N} \right\rbrack \right)} \middle| C_{\min}^{b} \right.} \right)} \leq \delta_{Th}^{b}}} & \left\lbrack {{Equation}\mspace{14mu} 52} \right\rbrack \\{{i_{\hat{e}} = {\max\limits_{i \in {\lbrack{i_{1}:i_{N}}\rbrack}}{i\mspace{14mu}{subject}\mspace{14mu}{to}}}}\mspace{14mu}{{I\left( {U_{i};\left. {\mathcal{L}_{i}^{e}\left( \left\lbrack {1:N} \right\rbrack \right)} \middle| C_{\max}^{e} \right.} \right)} \geq \delta_{Th}^{e}}} & \;\end{matrix}$

Herein, L_(i)([1:N]) denotes an LLR value for an i^(th) bit channel of amother code having a length N. In order for Bob to decode data withouterror, it must be determined as δ_(Th) ^(b)≃1. In order for Eve not todecode data, it must be determined as δ_(Th) ^(e)≃0.

FIG. 14 shows an example of values i_({circumflex over (b)}) and i_(ê)for a mother code.

The values i_({circumflex over (b)}) and i_(ê) of FIG. 14 are for apolar code having a length N and used as a mother code. It is assumedthat N coded bits in the mother code are divided into J groups, and eachgroup has a magnitude M_(j), j=1, 2, 3, . . . , J.|

₁ |=M ₁,|

₂ |=M ₂, . . . ,|

_(J) |=M _(J)  [Equation 53]

In this case, Σ_(j=1) ^(J) M_(j)=N, and

_(i) ₁ ∩

_(i) ₂ =θ, l₁≠l₂.

4.1 Secure HARQ scheme 1A

For a secure HARQ scheme 1A, a non-secure HARQ scheme 1 is extendedlyapplied to physical layer security communication. In the secure HARQscheme 1A, each group is configured as follows.

𝒢 j = { 1 ⋃ ℐ ⋃ ℛ , j = 1 j = 2 , 3 , … ⁢ , J [ Equation ⁢ ⁢ 54 ]

That is, a first group includes all information bits and all randombits. In addition, frozen bits belonging to Z₁ are also included in thefirst group. Except for the first group, the remaining all groupsconsist of only frozen bits.

For the first group, Z₁ may be determined optimally as follows.

1 opt = arg ⁢ ⁢ ⁢ ∑ i ∈ ℐ ⁢ ( I ⁡ ( U i ; ℒ i b ⁡ ( 𝒢 1 ) ) - I ⁡ ( U i ; ℒ i e⁡( 𝒢 1 ) | C max e ) ) + ⁢ ⁢ subject ⁢ ⁢ to ⁢ ⁢ I ⁡ ( U i ; ℒ i e ⁡ ( 𝒢 1 ) | Cmax e ) < δ Th e , ∀ i ∈ ℐ ⁢ [ Equation ⁢ ⁢ 55 ]

Herein,

, (

_(i)) denotes LLR of an i^(th) bit channel in a polar code having alength M₁ and consisting of only bits in the first group.

For a second group, Z₂ may be determined optimally as follows.

⁢2 opt = arg ⁢ ⁢ ∑ i ∈ ℐ ⁢ ( I ⁡ ( U i ; ℒ i b ⁡ ( 𝒢 1 opt ⋃ 2 ) ) - I ⁡ ( U i; ℒ i e ⁡ ( 𝒢 1 opt ⋃ 2 ) | C max e ) ) + [ Equation ⁢ ⁢ 56 ] subject ⁢ ⁢ to ⁢⁢I ⁡ ( U i ; ℒ i e ⁡ ( 𝒢 1 opt ⋃ 2 ) | C max e ) < δ Th e , ∀ i ∈ ⁢ I

Finally, Z_(J) of a J^(th) group may be determined optimally as follows.

J opt = arg ⁢ ⁢ ⁢   ∑ i ∈ ℐ ⁢ ( I ⁡ ( U i ; ℒ i b ⁡ ( ⋃ j = 1 J - 1 ⁢ 𝒢 j opt ⋃J ) ) -   I ⁡ ( U i ; ℒ i e ⁡ ( ⋃ j = 1 J - 1 ⁢ 𝒢 j opt ⋃ j ) | C max e )) + [ Equation ⁢ ⁢ 57 ] subject ⁢ ⁢ to ⁢ ⁢ I ⁡ ( U i ; ℒ i e ⁡ ( ⋃ j = 1 J - 1 ⁢𝒢 j opt ⋃ J ) | C max e ) < δ th e , ⁢ ∀ i ∈ ℐ

Complexity required to perform the optimization varies depending on thevalue M_(j), and when a code length is long, the required complexity maybe significantly high. To decrease the complexity, a simplified secureHARQ scheme is described.

4.2 Simplified Secure HARQ Scheme 1A

The simplified secure HARQ scheme 1A is conceptually similar to theaforementioned simplified non-secure HARQ scheme 1. However, in thesimplified secure HARQ scheme 1A, all information bits and all randombits are transmitted in a first group.

The simplified secure HARQ scheme 1A is constructed by sequentiallyselecting one output bit channel which provides maximum mutualinformation for each input bit.

FIG. 15 shows an example of transmission according to the simplifiedsecure HARQ scheme 1A. In addition, FIG. 16 is a conceptual view forexplaining concept of the simplified secure HARQ scheme 1A.

A first group may be configured as follows.

$\begin{matrix}{{\mathcal{G}_{1} = \underset{M_{1} - K - {{({N - \hat{e} + 1})}\mspace{14mu}{zeros}}}{\underset{︸}{\left\lbrack {i_{\beta_{1}},{i_{\beta_{1}} + 1},\ldots\mspace{14mu},i_{\hat{b}}} \right.}}},} & \left\lbrack {{Equation}\mspace{14mu} 58} \right\rbrack \\{\underset{K\mspace{14mu}{info}\mspace{14mu}{bits}}{\underset{︸}{i_{\hat{b} + 1},i_{\hat{b} + 2},\ldots\mspace{14mu},i_{\hat{e} - 1}}},\underset{{all}\mspace{14mu}{({N - \hat{e} + 1})}\mspace{14mu}{random}\mspace{14mu}{bits}}{\underset{︸}{\left. {i_{\hat{e}},i_{\hat{e} + 1},\ldots\mspace{14mu},i_{N}} \right\rbrack}}} & \;\end{matrix}$

Herein, β₁{circumflex over (b)}−(M₁−K−N+ê). If K+(N−ê+1) is greater thanM₁, security communication is impossible. In this case, the value M₁must be increased.

A second group may be configured as follows.

$\begin{matrix}{\mathcal{G}_{2} = \left\lbrack \underset{M_{2}\mspace{14mu}{zeros}}{\underset{︸}{i_{\beta_{2}},i_{\beta_{2 + 1}},\ldots\mspace{14mu},i_{\beta_{1} - 1}}} \right\rbrack} & \left\lbrack {{Equation}\mspace{14mu} 59} \right\rbrack\end{matrix}$

Herein, β₂=β₁−M₂.

By repeating the operation, a last J^(th) group may be configured asfollows.

$\begin{matrix}{\mathcal{G}_{J} = \left\lbrack \underset{M_{J}\mspace{14mu}{zeros}}{\underset{︸}{i_{1},i_{2},\ldots\mspace{14mu},i_{\beta_{j - 1} - 1}}} \right\rbrack} & \left\lbrack {{Equation}\mspace{14mu} 60} \right\rbrack\end{matrix}$

An example and concept for the secure HARQ scheme 1A described accordingto Equations 58 to 60 are as shown in FIG. 15 and FIG. 16.

4.3 Secure HARQ Scheme 1B

In the secure HARQ scheme 1B, each bit group is given as follows.

𝒢 j = { 1 ⋃ ℐ ⋃ ℛ 1 , j = 1 j ⋃ ℛ j , j = 2 , 3 , … ⁢ , J [ Equation ⁢ ⁢ 61]

Comparing with the aforementioned secure HARQ scheme 1A, in the secureHARQ scheme 1B, random bits are not all included in a first group, butare included in each group by a minimum amount necessary for securitycommunication.

Therefore, for the first group, Z₁ and R₁ may be determined as follows.

$\begin{matrix} & \left\lbrack {{Equation}\mspace{14mu} 62} \right\rbrack\end{matrix}$

For a second group, Z₂ and R₂ may be determined as follows.

( 2 opt , ℛ 2 opt ) = arg ⁢ max 2 , ℛ 2 ⊆ [ 1 : N ] ⁢ \ ⁢ 𝒢 1 opt ⁢ ⁢ ∑ i ∈ ℐ⁢( I ( U i ; ℒ i b ⁡ ( 𝒢 1 opt ⋃ 𝒢 2 ) ) - I ⁡ ( U i ; ℒ i e ⁡ ( 𝒢 1 opt ⋃ 𝒢2 ) | C max e ) ) + ⁢ s ⁢ ubject ⁢ ⁢ to ⁢ ⁢ I ⁡ ( U i ; ℒ i e ⁡ ( 𝒢 1 opt ⋃ 2 )| C max e ) < δ Th e , ∀ i ∈ ℐ [ Equation ⁢ ⁢ 63 ]

Finally, Z_(J) and R_(J) of the J^(th) group may be determined asfollows.

( J opt , ℛ J opt ) = arg ⁢ ⁢ max ⁢ J , ℛ j ⊆ [ 1 : N ] ⁢ \ ⁢ ⋃ j = 1 J - 1 ⁢𝒢 j opt ⁢ ∑ i ∈ ℐ ⁢ ( I ( U i ; ℒ i b ⁡ ( ⋃ j = 1 J - 1 ⁢ 𝒢 j opt ⋃ 𝒢 j )) -   I ⁡ ( U i ; ℒ i e ⁡ ( 𝒢 j opt ⋃ 𝒢 J ) | C max e ) ) + ⁢ ⁢ subject ⁢ ⁢ to⁢⁢I ⁡ ( U i ; ℒ i e ⁡ ( ⋃ j = 1 J - 1 ⁢ 𝒢 1 opt ⋃ J ) | C max e ) < δ Th e ,∀ i ∈ ℐ [ Equation ⁢ ⁢ 64 ]

In addition, a simplified secure HARQ scheme for decreasing complexityof the operations is described.

4.4 Simplified Secure HARQ Scheme 1B

The simplified secure HARQ scheme 1B is conceptually similar to theaforementioned simplified secure HARQ scheme 1A. However, all randombits are included in a first packet in the simplified secure HARQ scheme1A, whereas only minimum random bits required for security communicationare included in each packet in the simplified secure HARQ scheme 1B.

FIG. 17 shows an example of transmission according to the simplifiedsecure HARQ scheme 1B. In addition, FIG. 18 is a conceptual view forexplaining concept of the simplified secure HARQ scheme 1B.

A first group may be configured as follows.

$\begin{matrix}{{\mathcal{G}_{1} = \underset{M_{1} - K - {{({N - \alpha_{1} + 1})}\mspace{14mu}{zeros}}}{\underset{︸}{\left\lbrack {i_{\beta_{1}},i_{\beta_{1} + 1},\ldots\mspace{14mu},i_{\hat{b}}} \right.}}},\underset{{all}\mspace{14mu} K\mspace{14mu}{info}\mspace{14mu}{bits}}{\underset{︸}{i_{\hat{b} + 1},i_{\hat{b} + 2},\ldots\mspace{14mu},i_{\hat{e} - 1}}},\underset{{all}\mspace{14mu}{({N - \alpha_{1} + 1})}\mspace{14mu}{random}\mspace{14mu}{bits}}{\underset{︸}{\left. {i_{\alpha},i_{\alpha_{1} + 1},\ldots\mspace{14mu},i_{N}} \right\rbrack}}} & \left\lbrack {{Equation}\mspace{14mu} 65} \right\rbrack\end{matrix}$

Herein, β₁={circumflex over (b)}−(M₁−K−N+α₁), and the number i_(α1) ofminimum random bits for the first group required for securitycommunication may be determined as follows.

$\begin{matrix}{i_{\alpha_{1}} = {{\min\limits_{i \in {\lbrack{i_{\hat{e}}:i_{N}}\rbrack}}{i\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu}{I\left( {U_{i};\left. {\mathcal{L}_{i}^{e}\left( \mathcal{G}_{1} \right)} \middle| C_{\max}^{e} \right.} \right)}}} \geq \delta_{Th}^{e}}} & \left\lbrack {{Equation}\mspace{14mu} 66} \right\rbrack\end{matrix}$

If K+(N−α₁+1) is greater than M₁, security communication is impossible.In this case, the value M₁ must be increased.

A second group may be configured as follows.

$\begin{matrix}{{\mathcal{G}_{2} = \underset{M_{2} - {{({\alpha_{1} - \alpha_{2}})}\mspace{14mu}{zeros}}}{\underset{︸}{\left\lbrack {i_{\beta_{2}},i_{\beta_{2} + 1},\ldots\mspace{14mu},i_{\beta_{1} - 1}} \right.}}},\underset{{({\alpha_{1} - \alpha_{2}})}\mspace{14mu}{random}\mspace{14mu}{bits}}{\underset{︸}{\left. {i_{\alpha_{2}},i_{\alpha_{2} + 1},\ldots\mspace{14mu},i_{\alpha_{1} - 1}} \right\rbrack}}} & \left\lbrack {{Equation}\mspace{14mu} 67} \right\rbrack\end{matrix}$

Herein, β₂=β₁−(M₂−α₁+α₂). The number i_(α2) of minimum random bits forthe second group required for security communication may be determinedas follows.

$\begin{matrix}{i_{\alpha_{2}} = {{\min\limits_{i \in {\lbrack{i_{\hat{e}}:i_{\alpha_{1} - 1}}\rbrack}}{i\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu}{I\left( {U_{i};\left. {\mathcal{L}_{i}^{e}\left( {\mathcal{G}_{1}\bigcup\mathcal{G}_{2}} \right)} \middle| C_{\max}^{e} \right.} \right)}}} \geq \delta_{Th}^{e}}} & \left\lbrack {{Equation}\mspace{14mu} 68} \right\rbrack\end{matrix}$

By repeating the operation, a last J^(th) group may be configured asfollows.

$\begin{matrix}{{\mathcal{G}_{J} = \underset{M_{J} - {{({\alpha_{J - 1} - \alpha_{J}})}\mspace{14mu}{zeros}}}{\underset{︸}{\left\lbrack {i_{1},i_{2},\ldots\mspace{14mu},i_{\beta_{J - 1} - 1}} \right.}}},\underset{{({\alpha_{J - 1} - \alpha_{J}})}\mspace{14mu}{random}\mspace{14mu}{bits}}{\underset{︸}{\left. {i_{\alpha_{j}},i_{\alpha_{J} + 1},\ldots\mspace{14mu},i_{\alpha_{J - 1} - 1}} \right\rbrack}}} & \left\lbrack {{Equation}\mspace{14mu} 69} \right\rbrack\end{matrix}$

The number i_(αJ) of minimum random bits for the J^(th) group requiredfor security communication may be determined as follows.

$\begin{matrix}{i_{\alpha_{J}} = {{\arg{\min\limits_{i \in {\lbrack{i_{\hat{e}}:i_{\alpha_{J - 1} - 1}}\rbrack}}{i\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu}{I\left( {U_{i};\left. {\mathcal{L}_{i}^{e}\left( {\bigcup_{j = 1}^{J}\mathcal{G}_{j}} \right)} \middle| C_{\max}^{e} \right.} \right)}}}} \geq \delta_{Th}^{e}}} & \left\lbrack {{Equation}\mspace{14mu} 70} \right\rbrack\end{matrix}$

An example and concept for the secure HARQ scheme 1A described accordingto Equations 65 to 70 are as shown in FIG. 17 and FIG. 18.

Complexity required in optimization for the aforementioned simplifiedsecure HARQ scheme 1B is not high, and performance of the simplifiedsecure HARQ scheme 1B is better than the simplified secure HARQ scheme1A. However, the simplified secure HARQ scheme 1A has lower complexitysince no optimization is required.

4.5 Secure HARQ Scheme 2

It is not possible to construct secure HARQ based on the aforementionednon-secure HARQ scheme 2. This is because the non-secure HARQ scheme 2improves reliability of existing information bits by transmitting anadditional packet after information bits and frozen bits are optimizedbased on a first packet. If the non-secure HARQ scheme 2 is extendedlyapplied to the secure HARQ scheme, reliability of the existinginformation bits is improved, and thus a decoding error probability maybe decreased from a viewpoint of Bob, whereas security is lowered from aviewpoint of Eve.

4.6 Secure HARQ Scheme 3

Secure HARQ may be constructed based on the aforementioned non-secureHARQ scheme 3. The non-secure HARQ scheme is similar to the non-secureHARQ scheme 1 in a sense that a mother code is constructed and theconstructed mother code is punctured. However, the non-secure HARQscheme 1 performs puncturing after locations of an information bit and afrozen bit are optimized, whereas the non-secure HARQ scheme 3 optimizesthe locations of the information bit and the frozen bit after performingpuncturing. Similarly thereto, the secure HARQ scheme 3 may be performedby optimizing the locations of the information bit and the frozen bitafter constructing the mother code and puncturing the constructed mothercode so as to be a code having a desired length.

FIG. 19 is a flowchart showing a method of performing HARQ by using apolar code having a random length according to an embodiment of thepresent invention.

Referring to FIG. 19, a transmitter (Alice) generates a mother codehaving a length N (S100). Specifically, the transmitter generates amother code including an information bit constituting data to betransmitted and a frozen bit irrelevant to data to be transmitted.

The transmitter generates a code having a random length M by puncturingthe mother code (S200). Specifically, the transmitter may calculatemutual information on the basis of a probability distribution of LLR forthe mother code, and may puncture the mother code so that a loss of thecalculated mutual information is decreased. In this case, theprobability distribution of LLR may be calculated by using Gaussianapproximation. In this case, a ratio (i.e., a transfer rate) of aninformation bit and frozen bit included in the code having the randomlength M may be determined by a previously received indication signal ormay be predetermined.

The transmitter optimizes the code having the random length M (S300).Specifically, the transmitter determines locations of an information bitand a frozen bit in the code having the random length M so that avariance of the mutual information for the information bit and thefrozen bit is maximized

In addition, the transmitter applies the locations of the optimizedinformation bit and frozen bit to the mother code (S400). In addition,the transmitter performs HARQ by using a packet generated by splittingthe mother code (S500). Specifically, the transmitter transmits a nextpacket upon receiving NACK from a receiver, and performs HARQ byconstructing a new mother code upon receiving ACK from the receiver.

The aforementioned embodiments of the present invention can beimplemented through various means. For example, the embodiments of thepresent invention can be implemented in hardware, firmware, software,combination of them, etc. Details thereof will be described withreference to the drawing.

FIG. 20 is a block diagram illustrating a wireless communication systemin which the present disclosure is implemented.

The transmitter (Alice) 200 includes a processor 201, a memory 202, anda radio frequency (RF) unit 203. The memory 202 is connected to theprocessor 201 to store various information for driving the processor201. The RF unit 203 is connected to the processor 201 to transmitand/receive a wireless signal. The processor 201 implements a suggestedfunction, procedure, and/or method. An operation of the base station 200according to the above embodiment may be implemented by the processor201.

The receiver (Bob) 100 includes a processor 101, a memory 102, and an RFunit 103. The memory 102 is connected to the processor 101 to storevarious information for driving the processor 101. The RF unit 103 isconnected to the processor 101 to transmit and/receive a wirelesssignal. The processor 101 implements a suggested function, procedure,and/or method. An operation of the wireless 100 according to the aboveembodiment may be implemented by the processor 101.

A processor may include an application-specific integrated circuit(ASIC), another chipset, a logic circuit, and/or a data processor. Amemory may include read-only memory (ROM), random access memory (RAM), aflash memory, a memory card, a storage medium, and/or other storagedevices. An RF unit may include a baseband circuit to process an RFsignal. When the embodiment is implemented, the above scheme may beimplemented by a module (procedure, function, and the like) to performthe above function. The module is stored in the memory and may beimplemented by the processor. The memory may be located inside oroutside the processor, and may be connected to the processor throughvarious known means.

In the above exemplary system, although methods are described based on aflowchart including a series of steps or blocks, the present inventionis limited to an order of the steps. Some steps may be generated in theorder different from or simultaneously with the above other steps.Further, it is well known to those skilled in the art that the stepsincluded in the flowchart are not exclusive but include other steps orone or more steps in the flowchart may be eliminated without exerting aninfluence on a scope of the present invention.

What is claimed is:
 1. A method of performing a hybrid automatic repeatrequest (HARQ) based on physical layer security, the method comprising:generating a second code having a length different from that of a firstcode by puncturing the first code, the second code comprising aninformation bit constituting data to be transmitted and anon-information bit irrelevant to the data to be transmitted;determining a location of the information bit and a location of thenon-information bit in the second code; applying the determinedlocations of the information bit and non-information bit determinedbased on the second code to the first code; and performing the HARQ byusing a packet generated by splitting the first code, wherein in thegenerating of the second code, mutual information is calculated based ona probability distribution of a log likelihood ratio (LLR) for the firstcode, and the first code is punctured to decrease a loss of thecalculated mutual information, wherein the probability distribution ofthe LLR is calculated by using Gaussian approximation, and wherein aratio of the information bit and non-information bit in the second codeis determined by an indication signal previously received.
 2. The methodof claim 1, wherein in the determining of the locations of theinformation bit and the non-information bit, the locations of theinformation bit and the non-information bit are determined to minimize avariance of mutual information for each of the information bit and thenon-information bit.
 3. An apparatus for performing a hybrid automaticrepeat request (HARQ) based on physical layer security, the apparatuscomprising: a transmitter and a receiver capable of transmitting andreceiving a radio signal, respectfully; and a processor controlling thetransmitter and receiver, wherein the processor is configured to:generate a second code having a length different from that of a firstcode by puncturing the first code, the second code comprising aninformation bit constituting data to be transmitted and anon-information bit irrelevant to the data to be transmitted; determinea location of the information bit and a location of the non-informationbit in the second code; apply the determined locations of theinformation bit and non-information bit determined based on the secondcode to the first code; and perform the HARQ by using a packet generatedby splitting the first code, wherein the processor is configured tocalculate mutual information on the basis of a probability distributionof a log likelihood ratio (LLR) for the first code, and puncture thefirst code to decrease a loss of the calculated mutual information,wherein the probability distribution of the LLR is calculated by usingGaussian approximation, and wherein a ratio of the information bit andnon-information bit in the second code is determined by an indicationsignal previously received.
 4. The apparatus of claim 3, wherein theprocessor is configured to determine the locations of the informationbit and the non-information bit to minimize a variance of mutualinformation for each of the information bit and the non-information bit.